This is an inquiry into a fundamental
problem that has to do with the tropical versus sidereal zodiac controversy
and how it pertains to the practice of ancient astrology. It involves the
very ancient practice of granting periods of years to the signs of the
zodiac according to rising times. First of all let us define what is meant
by the term "rising times."

As each sign rises in the east a certain number
of degrees of right ascension pass over the meridian. This number of degrees
constitutes the "rising times" of the sign. A period was assigned to each
sign at the rate of one degree per year. For example at 40° north the
tropical sign Aries rises while 18°06' pass over the meridian. Taurus
rises while 21°47' pass over the meridian, etc.

For the technically inclined another way of defining
rising times is to compute the oblique ascension of the beginning and end
of each sign and to subtract the oblique ascension or O.A. of 0 degrees
of the sign from the O.A. of 30 degrees of that same sign. Oblique ascensions
will be discussed below.

The computation of rising times has a great antiquity.
According to Neugebauer in the History of Ancient Mathematical Astronomy
(HAMA) the Babylonians in an early system of computations known as "System
A" created such a scheme of rising times, although the times were computed
not according to modern trigonometric methods as described above but by
a numerical series such as that which follows: Starting with the first
30 degree segment of the zodiac which got 20 degrees of ascension, each
30 degree segment was given exactly 4 degrees more of rising than the preceding
segment up to the sixth segment. The sixth and seventh segments had the
same rising times, and subsequent segments decrease at the rate of 4 degrees
until the twelfth segment has the same rising time as the first. The longest
rising times of the sixth and seventh 30 degree segment were each 40 degrees.
We call these 30 degree segments rather than signs because according to
Neugebauer these 30 degree segments began not at 0 Aries but at 10 degrees
of Aries which was defined as being the location of the vernal point in
the zodiacal signs. While it is tempting to assume that what we have here
is measure of the vernal point in terms of a sidereal zodiac, the problem
is that scholars do not agree that this vernal point was regarded as moving.
The Babylonians may very well have regarded it as the permanent location
of the vernal point in the zodiac. For the most part scholars do not believe
that the Babylonians at this stage were aware that the vernal point precessed.
According to these scholars Hipparchos the Greek discovered precession
centuries after the development of System A. Other scholars such as Cyril
Fagan and others who for the most part would not be regarded as "mainstream"
[not necessarily to be interpreted as a criticism on my part], do believe
that the Babylonians knew about precession.

Pursuant to this discussion let us acknowledge
one thing at this point in the discussion. If we take Neugebauer's account
of these rising times at face value, then we must accept that at some level
that the Babylonians who developed this system were aware of the distinction
between these 30 degree "signs" and 30 degree segments measured from the
vernal point. It is not quite so clear that the Babylonians were conscious
"Siderealists." However, whether or not the Babylonians who developed System
A were conscious "Siderealists" is not central to our discussion.

Below is a table of the rising times of the 30
degree divisions from the vernal point as given in the Babylonian System
A.

The table is taken more or less directly from
Neugebauer's HAMA with slight modifications in the notation. The first
column simply refers to the 12 segments. The second column marked "R" contains
the rising times as defined above of each of these 30 degree segments of
the ecliptic. The third column gives a running total of the rising times
computed from the rising of the vernal point. The fourth column gives the
longitude in the zodiac in use (presumably a sidereal one) of the beginning
of each 30 degree segment measured from the vernal point at 10 degrees
of Aries. The fifth column gives the length of daylight that occurs when
then Sun is in the particular degree. Note that when the Sun is at the
vernal and autumnal points the daylight is precisely 12 hours; the daylight
reaches a maximum of 14h 24m when the Sun is exactly 90 degrees later at
the summer solstice point, and a minimum of 9h 36m when the Sun is at winter
solstice point. Also note that the ratio of the longest daylight period
to the shortest daylight period is precisely 3:2. Latitudes in the ancient
world were measured according to the ratio of longest day to shortest day.

Another school of Babylonians developed a second
system of rising times which are associated with a "System B." In this
system the 30 degree divisions are made from a vernal point which is defined
as 8 degrees of Aries. This is apparently a later measurement of the vernal
point in the sidereal zodiac. Below is a table of rising times. The values
are different from the previous table, but the logic of the table is the
same.

We include this table of System B for the sake
of completeness, but it will not figure in our discussion as much as System
A except in one regard, the definition of the vernal point as being 8 degrees
of Aries.

Note that in both tables opposite 30 degree segments
have rising times which total 60 degrees. Also note the following: the
first and twelfth segments have the same rising times; so do the second
and eleventh segments, the third and tenth segments, and so forth. This
symmetry is very important for reasons that we will disclose below. First
however let us look at how well the rising times of System A reflect the
actual rising times of the 30 degree segments as computed according to
modern methods.

Rising Times of the Tropical Signs Using Modern
Values Versus the Values of System A

The last row is computed for 501 B.C.E. using modern
trigonometric methods for Babylon. The agreement is quite good for R1-R12,
R2-R11, and R5-R8. The fit is cruder for the other segments. It is also
worth noting that the rising times computed for these 30 degree segments
measured from the vernal point (which of course correspond to the tropical
signs) are quite stable over time. Below is a table showing the rising
times computed using modern methods for 501 B.C.E. and 2000 C.E. for the
Babylon.

The reader will note the differences are small. For
all practical purposes the rising times of the tropical signs for any particular
latitude are virtually constant over time.

By contrast let us look at the rising times of
sidereal signs using the Fagan-Allan ayanamsha [the difference between
the tropical and sidereal zodiacs] computed for the same two epochs again
using modern methods.

Upon examining the table two things become apparent.
First of all the rising times of the sidereal signs are far from constant
over time. Note particularly the rising time of sidereal Gemini in 501
B.C.E. which equals 31°42'. But in the year 2000 C.E. Gemini's rising
times will be 24°42' a difference of 7 degrees!

Second the symmetry that we noted between pairs
of signs R1-R12 (Ar-Pi), R2-R11 (Ta-Aq), R3-R10 (Ge-Cp), etc. is not present.
In 501 B.C.E. Gemini gets 31d 42m while the sign that ought to be symmetrical
with it, Capricorn, gets 27d 33m. In the year 2000 C.E. Gemini gets 24d
42m and Capricorn gets 34d 04m. This is because the symmetry of the rising
times of the 30 degree segments requires that the measuring of the segments
be done from one of the equinoctial points. This may require some explanation.

Coordinate Table for the Beginnings of the Tropical
Signs for 40 Degrees North Latitude

In the table given above we have the following: The
first row marked "Long." contains the tropical longitudes of the beginning
of each sign. The second row marked "R.A." contains the right ascension
of the beginning of each sign. The row marked "Decl." is declination of
the ecliptic degree at the beginning of each sign. The row marked "A.D."
contains the ascensional difference of the beginning of each sign. This
will be explained shortly. The row marked "O.A." contains the oblique ascension
of the beginning of each sign. This will also be explained shortly. And
last the row marked "R.T." contains the rising times of the signs which
begin at the designated longitude.

First an explanation of oblique ascension. On
the equator all positions on the celestial sphere, regardless of declination,
rise along with their right ascensions at 0 degrees declination. This is
because at the terrestrial equator the celestial equator rises in the east
exactly perpendicular to the horizon, hence the term "right" ascension,
"right" meaning perfectly upright. But either north or south of the terrestrial
equator positions on the celestial sphere do not rise with their positions
measured in right ascension. They rise along some other degree on the celestial
equator. This other degree is the oblique (or slantwise) ascension of our
hypothetical position on the celestial sphere. It is called oblique ascension
because the celestial equator at latitudes other than 0 degrees north or
south rises slantwise or obliquely in the east, the further away from 0
terrestrial latitude (the equator), the more obliquely. Therefore, the
oblique ascension of position A can be defined as whatever degree on the
equator may be rising when A exactly touches the horizon assuming that
A is not on the celestial equator, i.e., that A has a declination not equal
to 0.

Ascensional difference or A.D. is a measure of
the difference between the R.A. or right ascension of a point and its O.A.
or oblique ascension. The formula is as follows:

O.A. = R.A. - A.D.

Thus the A.D. of a point is required to find the
O.A. of that point. The A.D. of a point in turn is derived from the declination
of the point and the terrestrial latitude of the place in question by the
following formula.

A.D. = arcsin(tan Decl. x tan Latitude)

These relationships can be seen in the coordinate
table shown above. Note from the table that arc from the O.A. of 330 degrees
to the O.A. of 360 or 0 degrees is the same as the arc from the O.A. of
0 degrees to the O.A. of 30 degrees. These arcs are the rising times of
tropical Pisces and Aries. This is the consequence of the following two
facts: First that the arc in R.A. from 0 degrees tropical Pisces to 0 degrees
tropical Aries is the same as the arc in R.A. from 0 degrees tropical Aries
to 30 degrees tropical Aries (or 0 degrees Taurus). The second fact is
that the declination of 0 degrees tropical Pisces is the exact opposite
of the declination of 30 degrees tropical Aries, the first being -11.477
degrees, the second being +11.477. This in turn causes the A.D. of 0 degrees
tropical Pisces to be the exact opposite of the A.D. of 30 degrees tropical
Aries.

Only if two points are symmetrical with respect
to the equinoxes can they possess this symmetry of arcs in O.A. which in
turn produces the symmetrical rising times of signs which are equidistant
from the equinoxes. This symmetry can occur in a sidereal zodiac only when
the vernal point is at exactly 0 degrees of a sign. The Babylonians of
Systems A and B knew this which is why they measured the rising times of
30 degree arcs from the vernal point rather than from 0 degrees of Aries
in a zodiac in which the vernal point was not at 0 Aries (or any other
sign).

This tells us something very important which seems
to have escaped the notice of nearly everyone. The Babylonians of Systems
A and B had at least two twelvefold divisions of the ecliptic into 30 degree
divisions: One was made from a point which was 10 or 8 degrees prior to
the vernal point. This "zodiac" may or may not have been consciously sidereal.
The second was a "zodiac" which was measured from the vernal point and
which clearly was consciously tropical. However, was this "tropical zodiac"
actually used for any astrological purpose? For that matter was the other,
possibly consciously sidereal, zodiac used for astrological purposes? The
usual answer to the latter question is yes, but this is a matter which
we will need to examine further. Likewise the answer to the first question
is usually said to be no. But the fact remains that there was a twelvefold
equal tropical division of the ecliptic in Babylonian times along with
the probably sidereal one. This paves the way for the next stage of things.

The central difficulty that has not been dealt
with in this controversy is this question. What was the practice of astrologers
when they came to do astrology more or less as we know it? What zodiac
did they use, and were they really conscious of what they were doing? For
it is not enough to show that early charts were computed using a sidereal
zodiac if the people who cast them were not aware that they were using
a sidereal zodiac. And the entire controversy becomes moot if it can be
shown that for all practical purposes the two zodiacs were not distinguished.

The oldest known birthchart has been dated by
Sachs to 410 B.C.E. It is a cuneiform chart with no degrees given, only
sign positions, also no Ascending degree. Computations using both tropical
and sidereal zodiacs give the same signs for all of the planets so listed.
This chart therefore is of no use in determining the zodiac in use. The
next several charts in cuneiform date from the 3rd century B.C.E. These
do contain degrees for individual planets and these positions are reasonably
consistent with positions in the Fagan-Allen sidereal zodiac. However,
they could also conceivably be computed in a the zodiac in which the vernal
point is fixed at 8 Aries, the System B zodiac. None of these charts have
Ascendants which means that if these charts are accurate exemplars of the
chart technology of the age, we are dealing with a very primitive form
of horoscopy, not the sophisticated one that appears in later Greek astrology.
Yet we are well within the period that appears to be the date for the Nechepso-Petosiris
text which already shows an advanced horoscopic technique. Could Egyptian
horoscopy have already outstripped the Babylonian art on which it was undoubtedly
based?

The following table shows the close coinciding
of the zodiacs through the period in which horoscopic astrology comes into
being and begins truly to flower. The longitudes are the positions of the
vernal point given in terms of the sidereal zodiac of Fagan-Allen. It is
clear that only the most precise astronomical computations could allow
us to clearly distinguish the two zodiacs in the main period of Greek astrology.

So do we see evidence of the astrologers of this
period having a clear idea of their zodiac? Of course there is the example
of Ptolemy (roughly 175 C.E.) who explicitly states that the zodiac begins
with the vernal point. And we also have his witness that Hipparchos supported
the same position about 300 years before. But was this issue clearly delineated
in the minds of astrologers in general? That is the question.

Let's explore the answer to this question first
of all by means of the ascensional times which we have been describing
here. When astrology clearly emerges into the documentable daylight, the
early centuries C.E. (A.D.), the Babylonian doctrine of ascensional times
comes along with it.

Vettius Valens was a younger contemporary of Ptolemy,
about 175 C.E., but his astrological style is clearly derived from a different
tradition than Ptolemy's, one at least as old. In the Anthology, Book I,
chapter 6, there is the following passage.

"One must know how much addition or subtraction
of the ascension each zoidion has, thusly. Since Aries ascends in 20 times,
Libra ascends in 40 in order to fill up the 60 times. For, in comparison
to the number of times in which each zoidion ascends, the zoidion diametrically
opposite takes a number that fills up 60 times. And in comparison to the
number of hours for each zoidion, the hours for the zoidion diametrically
opposite fill up 4 hours. And in comparison to the number of days and months,
the zoidion opposite takes a number that fills up two years. For, the amount
by which each zoidion exceeds, the diametrically opposite zoidion is lacking."Subtract, then, the least from the present greatest,
that is, the 20 times from the 40. The remainder is 20. 1/5 of these becomes
4. The addition or subtraction of each zoidion is four. If, then, we add
4 to the 20 ascensions, 24 ascensions result. Taurus will ascend in these
times. Gemini in 28, Cancer in 32, Leo in 36, Virgo in 40, Libra in 40.
Then similarly, from Scorpio you subtract 4 up to Pisces. Inquiring thus,
you will get to know the ascensions for each zone." (Schmidt Trans.)

These are the standard System A ascensional times.
However, as we know from Books VIII and IX, even though Valens used System
A ascensional times, he set the vernal point at 8 degrees of Aries, the
position used in System B. And in addition he and the others who used this
system did not make the distinction that the Babylonians seemed to have
made. He did not locate the 30 degree tropical divisions which gave rise
to the ascensional times at 8 degrees of the signs but at 0 degrees of
the signs thus identifying the signs of the zodiac with the divisions that
gave rise to the ascensional times. And there is no evidence that this
was an innovation of Valens, but rather a common convention among the astrologers
of the period. This clearly indicates that these astrologers did not have
a clear idea of the distinction between sidereal and tropical zodiacs,
or that they did not consider the distinction to be important, not a remarkable
position given the fact the two zodiacs did nearly coincide at the time.

Also characteristic of the texts of the time are
references to the signs in terms that we would clearly recognize as based
on seasonal criteria along with factors that we clearly recognize as sidereal.
Consider the following description from Valens Book I. (Robert Schmidt
Trans.)

"Aries is the house of Ares, a masculine
zoidion, tropical, terrestrial, authoritative, fiery, free, ascending,
semi-vocal, good, changeable, administrative, public, civic, unprolific,
servile, Midheaven of the cosmos and cause of repute, two-colored (since
the Sun and the Moon make leprosies), skin-eruptions; it is also unconnected,
a place for eclipses. . . ."". . . This zoidion has 19 bright stars. And
it has 13 bright stars through the belt, 27 shadowy ones, 28 underbright
bright ones, and 48 faint ones. The first part of Perseus from the northern
regions co-rises with it, as does the remaining parts and the left part
of Auriga, and from the south the back fin and tail of the Sea Monster.
From the north, the [feet] of Bootes set. From the south, the remainder
of Lupus."

Note the use of the word "tropical" in the first
paragraph, followed by the enumerations of the stars in the second. [Several
paragraphs have been omitted.] This kind of thing is typical of the Greek
astrologers. Only Ptolemy seems to have been aware of the impending difficulty
and made an effort to differentiate clearly.

According to Neugebauer there were others who
adopted the vernal point at 0 degrees Aries. From page 600 of HAMA.

"We know from Hipparchus that the majority
of old mathematicians divided the ecliptic in this form. This statement
agrees with sources still available to us; Euctemon (about -430) placed
all four cardinal points on the first day of the respective signs. The
same holds for Callipus (about -330) and is underlying the era of Dionysius
(beginning -284/3). As far as we know this norm is attested nowhere in
Babylonian astronomy."

This last statement is somewhat weakened by the facts
given earlier, but I think that it can be taken as correct as far as stellar
and planetary positional measurement is concerned. It is not so clear as
far as astrological purposes are concerned.

When did the issue of precession become clear
to the astrologers? Clearly some sources were aware that stars did not
stay in the same place. In the second star list of the Liber Hermetis we
have the following passages on the degree of Cancer in reference to Praesaepe.
(All passages below translated by the author.)

"From the fourth to the seventh degree
according to the Sphaera Barbarica is the Little Cloud, . . ."

"From the seventh degree to the eighth there rises
Praesaepe and Lyra playing the lyre. But indeed there are those who say
that the Little Cloud should be in the eighth and ninth degree."

"In the ninth degree there rises the Little Cloud
of Cancer according to Dorotheus."

"From the eleventh to the twelfth degree is the
Little Cloud."

"From the twelfth degree and six minutes to the
fourteenth degree and 49 minutes there rises the Little Cloud; but according
to Ptolemy it is in the thirteenth degree."

Then we have the evidence of The Anonymous of 379
who may seem to some to be an ambiguous source for reasons which we will
explain below. First of all let us look at his own statement on precession.

". . . in what remains we will begin
to state the effects concerning the active power of each of the non-wandering
stars, after indeed inscribing in the table the degree number in longitude
which each of them occupies in the consulship of Olybrius and Ausonius,
at which time we wrote this book. This is on account of the fact that the
non-wandering stars move 1 degree into the following tropical signs in
100 years, just as the divine Ptolemy exemplified." (Schmidt Trans.)

This is obviously an explicit reference to a tropical
system with the vernal point at 0 degrees Aries and explicitly based on
Ptolemy. However, there is a problem with the Anonymous' positions as given
in the text. For example, Aldebaran is given in the text as the 15th degree
of Taurus. This is exactly the position given in Cyril Fagan's reconstruction
of the sidereal zodiac. Similarly Antares is given as the 15th degree of
Scorpio. These and other such positions have led certain investigators
to assume that the Anonymous was a siderealist! However, if one corrects
the positions given in Ptolemy's star catalog using Ptolemy's inaccurate
precessional constant, as the Anonymous explicitly did, one gets exactly
the same position. The following table shows this.

The difference between the Ptolemaic positions and
those of the Anonymous is exactly the 2 degrees that one would get using
the erroneous value for precession of Ptolemy's. Only Regulus is different
in the Anonymous but the corrected Ptolemy position gives exactly the sidereal
position. The coinciding of the positions given by the Anonymous with the
sidereal positions is a coincidence! And other authors that are sometimes
cited as siderealists are from the same general period, Hephaistio of Thebes,
and Firmicus Maternus.

On page 10 of The Zodiac: A Historical Survey
by Robert Powell the author cites a passage from Neugebauer's HAMA as evidence
for the Anonymous being a siderealist. Unfortunately the passage in question
is one in which Neugebauer is dating this author and another author named
Cleomedes to the 4th century by showing that their values for star positions
are derived from correcting Ptolemy's positions using his precessional
constant! One wonders how much of the evidence for the sidereal zodiac
among the Greeks comes from similarly questionable research.

When did the confusion of the zodiacs clearly
end? There is no simple answer to this question. However, it is clear that
the kind of confusion that we are documenting here survived among the Hindus.
The following is from Varahamahira's Brihat Jataka, chapter 1, sloka 19.

"The measures of the first six signs are represented
by the numbers 20, 24, 28, 32, 36, and 40 respectively. The same figures
taken in inverse order give the measures of the second six signs."

These are our old friends the System A rising
times for Babylon again; and again, just as in Valens, they are identified
with the signs of the zodiac, not a separate set of 30 degree divisions
having no fixed relation to the signs of the zodiac. And again they are
symmetrical with respect to 0 degree Aries, something that can only happen
in a tropical zodiac. Was this eminent figure of the Hindu tradition a
tropicalist? Apparently so. In another early Hindu work, the Yavana Jataka,
we also find symmetrical rising times, indicating a tropical zodiac although
these rising times at least are recomputed for India.

We now have to deal with a fundamental question:
which zodiac would have seemed the more reasonable to the ancients? This
is not a trivial question because it has been argued by moderns that a
sidereal zodiac would seem on the face of it to be more rational. After
all while the stars do have a small motion relative to each other, their
proper motions, most of their apparent motion is in fact due to the backwards
motion of the vernal point with respect to the fixed stars. It is obviously
more rational from a modern point of view to consider the one point as
being in motion rather than the many points. Also modern astronomy regards
the solar system from the point of view of the Sun rather than the Earth,
and it is therefore more reasonable to regard the stars as more or less
stationary than it is to so regard one single Earth-related point, the
vernal point.

But these are the criteria of moderns. Would they
have been the criteria of the ancients? Certainly it would have been easier
for them to measure positions with regard to fixed stars and we have abundant
evidence of the practice. But we also know that various persons among the
ancients were in fact quite capable of locating the cardinal tropical points
as is evidenced by the number of allignments to the rising positions of
the these points all over Europe.

The problem is that most of the ancients regarded
the Earth as being completely stationary. There were exceptions such as
some Pythagoreans, Aristarchos (who actually posited a heliocentric theory),
and at least one Hindu, Aryabhata. The most common view was that there
were eight spheres surrounding the earth. The eighth sphere held the fixed
stars and also rotated about the stationary earth once every twenty-four
hours, what was later called the Primum Mobile. The other seven are the
spheres of the seven planets. Here is the description of the creation of
these spheres from the Timaeus of Plato.

"And thus the whole mixture out of which he cut
these portions was all exhausted by him. This entire compound he divided
lengthways into two parts, which he joined to one another at the centre
like the letter X, and bent them into a circular form, connecting them
with themselves and each other at the point opposite to their original
meeting-point; and, comprehending them in a uniform revolution upon the
same axis, he made the one the outer and the other the inner circle. Now
the motion of the outer circle he called the motion of the same, and the
motion of the inner circle the motion of the other or diverse. The motion
of the same he carried round by the side to the right, and the motion of
the diverse diagonally to the left. And he gave dominion to the motion
of the same and like, for that he left single and undivided; but the inner
motion he divided in six places and made seven unequal circles having their
intervals in ratios of two-and three, three of each, and bade the orbits
proceed in a direction opposite to one another; and three [Sun, Mercury,
Venus] he made to move with equal swiftness, and the remaining four [Moon,
Saturn, Mars, Jupiter] to move with unequal swiftness to the three and
to one another, but in due proportion.

The equatorial motion of the eighth sphere is
designated the circle of the same or invariant, while the other seven circles,
those of the planets, are derived from the circle "of the other or diverse."
Then, when precession became a clearly understood doctrine, to the eighth
sphere was added a ninth sphere which became the primum mobile, and the
old eighth sphere held the fixed stars. And these were perceived as moving
with respect to the sphere of the primum mobile. This fact clearly demonstrates
that the fixed stars were conceived as being in motion with respect to
the primum mobile; the components of the primum mobile are the vernal point,
the celestial equator, the other equinox, and the solstices.

Nor is this all. We have already shown that the
rising times in the tropical zodiac, a critical feature of the ancient
system, is nearly invariant over time while the rising times of the constellations
are not. And those who measured the positions of the Sun at dawn along
the horizon would have noticed that the Sun always rose at the same position
along the horizon at the same time of year and that the maximum northerly
and southerly positions along the horizon were virtually invariant. And
even in Babylonia we know that the first constellation of the stellar zodiac
was the one which rose at dawn in the spring. Clearly even they, insofar
as they knew that the stars and vernal point were moving with respect to
each other, would have regarded the stars as being in motion, not the vernal
point.

From what we have seen it is clear that the Babylonians
had two divisions of thirty degrees, one corresponding to the constellations
which they may or may not have known were moving with respect to the vernal
point, and another which was fixed with respect to the vernal point with
the vernal point at 0 degrees. The Greeks did not invent the tropical zodiac
as often charged. All they did was to give the names of the constellations
to the tropical signs. We actually have no way of knowing at this point
what the ancients regarded as the astrologically effective set of divisions,
or even if they did regard only one set as being effective. Later generations
right up to modern times in the West regarded both sets as being effective,
but the later medieval and renaissance astrologers did not regard the constellations
as being equal. They regarded them as asterisms of unequal extent.

To conclude: I do not assert that the ancients
were tropicalists, nor do I assert that they were siderealists. I assert
that whatever they may have known about precession they tended not to make
the distinction, and when they did, they would have been just as likely
to give precedence to the tropical as the sidereal for divinatory purposes.
After all the pictorial constellations were only physical plane images
which roughly corresponded to the ideal, mathematical reality which would
have been represented by the tropical system. But fundamentally I believe
we have to regard the tropical-sidereal controversy as yet another example
of a historical pseudo-problem created by anachronistically projecting
a modern problem with modern points of view back onto the ancients. It
was not a problem with which the ancients were seriously concerned. Given
the limits of their computational accuracy, both systems would have given
them the same results. This is a question that we have to solve for ourselves.
An appeal to history will not work.

Ed. N.: This article is published on the website of Robert Hand,
the ARHAT (Archives for the Retrieval of Historical Astrological Texts). Thanks
to him for his permission to publish it here.